Spatial Perception II: Bayes Rule

Question 1

Accidental view

Answer 1

very rare perspective of a scene that would change radically if one were to shift

Question 2

Generic view

Answer 2

frequent perspective of a scene that would not change much if one were to shift
* inverse optics takes this into account and usually guesses it is a generic view (three shapes occluding each other example)

Question 3

How is the unconscious inference achieved?

Answer 3

the nervous system calculates the probability of each scene given the sensory evidence AND prior knowledge, then chooses the scene that has the highest probability

aka Bayes Rule

Question 4

What two sources of information contribute to the process of perception

Answer 4

• • prior knowledge

- - information provided by the senses

Question 5

Bayes Rule

Answer 5

P(SX | I) ∝ P(I | SX)P(SX)

posterior ∝ likelihood * prior

Question 6

Inverse Optics

Answer 6

the fundamentally ambiguous mapping between sources of retinal stimulation and the retinal images that are caused by those sources

• • the brain is in the business of inference; figuring out what generated a scene
• • sensations are a GENERATIVE PROCESS governed by rules of physics

Question 7

What do we call the rules of physics for perceiving images?

Answer 7

Laws of Optics

Question 8

Why do we say “inverse optics”?

Answer 8

The brain must work the laws of optics backwards to figure out what created a retinal image
– evidence and prior knowledge

Question 9

Prior Knowledge

Answer 9

knowledge the brain knows about the brain independent and before the image is seen

example: pennies are usually the same size and a complete circle

Question 10

Retinal Image

Answer 10

pure sensory information received on the retina from viewing the scene

example: one penny appearing larger than the other

Question 11

Who created the term “unnoticed judgment”

Answer 11

Al-Haytham/Alhazen

• • the FIRST SCIENTIST, father of optics
• • contributed a lot to physics and perception science but largely unknown
• • lived in house arrest in Egypt

Question 12

Bayes Theorem

Answer 12

P(SX | I) ∝ P(I | SX)P(SX)

posterior ∝ likelihood * prior

Question 13

How would you say Bayes Theorem in a paragraph

Answer 13

• the probability of a given scene (A, B, or C) GIVEN that this retinal image has occurred
• is proportional to the probability of getting an image like that if there was that scene out there
• multiplied by the probability of that scene happening in the first place

Question 14

How would you say Bayes Theorem in direct mathematical words

Answer 14

• probability of a SCENE given a RETINAL IMAGE
  [is equal to… ]
• the probability of that IMAGE given that SCENE
  [times … ]
  *the probability of that scene divided by a constant

Question 15

Posterior probability

Answer 15

probability of this scene happening given AFTER we have made an observation receiving the retinal information

Question 16

Likelihood

Answer 16

how often is this retinal image received given a scene

– note: think accidental vs generic views (inverse optics)

Question 17

Prior probability

Answer 17

how likely it is to come across something given information the brain knows about the world BEFORE we have made an observation

Question 18

In summary, what are the different variables of Bayes Theorem?

Answer 18

• sensory information (retinal)
• probability of getting this sensation given the scene i.e. how often it happens (accidental vs generic)
• prior knowledge about the world (expectation or bias)

Question 19

What is the constant “I”

Answer 19

probability of ALL possible retinal images

- remains the same across all scenes

Question 20

How does the brain use Bayes Theorem to perceive depth?

Answer 20

• each cue makes an estimate about depth
  P(d | c1,c2,c3,…)∝ P(c1 |d)P(c2 | d)P(c3 | d)…P(d)
  posterior ∝ likelihoods * prior

“probability of the distance of the object GIVEN that we have different cues with different information, WHAT IS THE PROBABILITY of a certain distance?”
note: assume different cues are independent of each other

e.g. probability of aerial view cue given the distance is 20 meters

Question 21

How does cue combination work to determine depth?

Answer 21

If P(d) is uniform, this will become equivalent to weighted averaging of the different cues, where each cue is weighted by its reliability

Question 22

What is a simple way to phrase the Bayes Theroem for depth in words

Answer 22

The optimal estimate of how far an object is = the weighted average of cues’ estimates X prior knowledge

*cues that are more reliable get a bigger weight